is it possible to make a reflection sit at infinity, like a reflex sight or a HUD?
As the wiki page said, you need your lens to refract parallel light.

Key Points


the lens need to be perfect refraction lens or it will cause distortion
the iron sight laser thing need to on the exatly same position of the lens focus point
the reflective plane should be 45 degree above the lens


Workflow

You can create a nearly perfect refraction lens by intersect two ball.


After the lens is setup, you need to know its focus length. It can be done several ways. I use a sun light with orthogonal view to tune the point on a plane:



You need to change parameter in render panel, for those thing will clamp indirect light, turn it off to get accurate result.

And last thing, your 45 degree reflective glass need to be fully transparent without distortion, recommend to use transparent shader + Glossy shader, and place above the lens correctly:



Then the result can be stable in sight with slightly distortion:



The full scene is extremely complicated in rendering, consider something else. 

And for the file I test for it in Blender 2.80: